The present invention is a method for rapidly computing the change in a modeled wavefield (i.e. pressure or displacement) given a small change in the model being considered. The method is applicable to forward modeling being performed in the frequency domain, wherein the wave equation has been formulated to propagate a wave at a single temporal frequency. As shown below, the frequency domain wave equation, known as the Helmholtz equation, becomes a set of linear equations after discretization in space.
The set of linear equations is typically solved in one of two ways, either via an iterative process or by factorizing the matrix representing the linear equations. (See, for example, reference 2—Shen Wang et al. (2010), which is incorporated herein by reference in all jurisdictions that allow it.) The invention to be described herein is most advantageous when applied in conjunction with the factorization method, which makes it possible to solve for the wavefield corresponding to an arbitrary source location very quickly once the factorization has been performed and the factors stored.
The primary disadvantage of the factorization method is the large amount of memory and long CPU time required to perform the matrix factorization. The elements of the matrix depend upon the model parameters, such as P-wave velocity Vp and the P-wave quality factor Qp. When these model parameters are modified, the matrix must be factorized again. Therefore, in an iterative process such as full wavefield inversion (“FWI”), where the model is changed many times, the computational cost is increased by a large amount due to these re-factorizations. Seismic data inversion is used for exploring for hydrocarbon reservoirs and for developing and producing them.